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Representing Measures and Hardy Spaces for the Infinite Polydisk Algebra
Author(s) -
Cole Brian J.,
Gamelin T. W.
Publication year - 1986
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-53.1.112
Subject(s) - mathematics , unit sphere , haar measure , measure (data warehouse) , hardy space , boundary (topology) , pure mathematics , unit (ring theory) , torus , combinatorics , discrete mathematics , algebra over a field , mathematical analysis , geometry , mathematics education , database , computer science
Representing measures and Jensen measures are studied for the uniform algebra on the infinite polydiskΔ ¯ ∞generated by the coordinate functions z 1 z 2 ,…. Let σ be the Haar measure on the infinite torus T ∞, which is the distinguished boundary of the infinite polydisk. For fixed p in the range 1 < p , < ∞, it is shown that a point ζ ∈ Δ∞ has a representing measure in L P (σ) if and only if ζ ∈ l 2 . A related result for a class of representing measures for the origin, including the Haar measure σ, and for fixed p in the range 0 < p < ∞, is that the point evaluation at ζ is continuous in the L p ‐norm if and only if ζ ∈ Δ∞ ∩ l 2 . In this case the functions in H p are shown to correspond to analytic functions on the domain Δ∞ l 2 in l 2 . Along these same lines, it is shown that H ∞ (σ) is isometrically isomorphic to H (Δ∞ ∩ l 2 ), and also to H ∞( B ), where B is the open unit ball of the sequence space c 0 .